Binomial Probability: The Likelihood of Two Drunks Meeting After N Steps

Assignment Task

1. Two drunks start out together at the origin, each having equal probability of making a step to the left or right along the x axis. Find the probability that they meet again after N steps. It is to be understood that the men make their steps simultaneously.

2. The probability W (n,) that an event characterized by a probability p occurs n, times in n trials was shown to be given by the binominal distribution

W(n) = n!/n!(n-n1)! p(1-p)^n-n1

Consider a situation where the probability p is small (p<<1>

(a) Using the result In (1-p)-p, show that (1-p)^{n-n1= e-np}

(b) Show that n(n-n)!≈n"

(c) Hence show that Eq. (1) reduces to

W(n1) = λm1/n1! e-λ

where np is the mean number of events. The distribution (2) is called the Poisson distribution.

3. In the 1-D molecular diffusion example, assume that p and q are the probabilities of the molecule moving to the right and left at each step and the molecule is at the position x = ml after n steps. If → 0 and n→ ∞, show that the probability P(x) is given by the standard form of Gaussian's distribution

P(x ) = 1/ 2Πσ e-(x-u)^2/2σ2

where σ = 2√npql and μ= (p-q)nl.

4. Consider the molecular diffusion example. After n steps,

(a) What is the mean displacement from the origin?

(b) What is the dispersion (x-x)2?

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